Using an old laptop with a old and depleted or soon to be depleted (especially when charged 100% constantly) wouldn't make me feel that comfortable, hardly something it was made for either.
The laptop itself is can be kind of neat as a home server, it usually has decent power consumption and if the screen works you don't have to dig out a keyboard and screen in case of an emergency. But you might want to leave the battery out, and this assuming that is is sensible to use the power brick 24/7. Which I wouldn't assume.
A more performant alternative is the Intel NUC, the coming i5 version also has hardware support for VNC alleviating the need for a screen/keyboard even for the most severe emergencies.